Is 7(cot^2(x))/(csc(x))sec^2(x)=7tan(x)cos(x)csc^2(x) ?

The answer to whether 7(cot^2(x))/(csc(x))sec^2(x)=7tan(x)cos(x)csc^2(x) is True

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prove 7(cot^2(x))/(csc(x))sec^2(x)=7tan(x)cos(x)csc^2(x)

Solution

prove

7 \frac{cot ^{2} ( x )}{csc ( x )} sec^{2} (x) = 7 tan (x) cos (x) csc^{2} (x)

Solution

True

Solution steps

7 \cdot \frac{cot ^{2} ( x )}{csc ( x )} sec^{2} (x) = 7 tan (x) cos (x) csc^{2} (x)

Manipulating left side

7 \frac{cot ^{2} ( x )}{csc ( x )} sec^{2} (x)

Express with sin, cos

7 \cdot \frac{cot ^{2} ( x )}{csc ( x )} sec^{2} (x)

Use the basic trigonometric identity:

cot (x) = \frac{cos ( x )}{sin ( x )}

= 7 \cdot \frac{( \frac{c o s ( x )}{s i n ( x )} ) ^{2}}{csc ( x )} sec^{2} (x)

Use the basic trigonometric identity:

csc (x) = \frac{1}{sin ( x )}

= 7 \cdot \frac{( \frac{c o s ( x )}{s i n ( x )} ) ^{2}}{\frac{1}{s i n ( x )}} sec^{2} (x)

Use the basic trigonometric identity:

sec (x) = \frac{1}{cos ( x )}

= 7 \cdot \frac{( \frac{c o s ( x )}{s i n ( x )} ) ^{2}}{\frac{1}{s i n ( x )}} (\frac{1}{cos ( x )})^{2}

Simplify

7 \cdot \frac{( \frac{c o s ( x )}{s i n ( x )} ) ^{2}}{\frac{1}{s i n ( x )}} (\frac{1}{cos ( x )})^{2} : \frac{7}{sin ( x )}

7 \cdot \frac{( \frac{c o s ( x )}{s i n ( x )} ) ^{2}}{\frac{1}{s i n ( x )}} (\frac{1}{cos ( x )})^{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{( \frac{c o s ( x )}{s i n ( x )} ) ^{2} \cdot 7 ( \frac{1}{c o s ( x )} ) ^{2}}{\frac{1}{s i n ( x )}}

Apply the fraction rule:

\frac{a}{\frac{b}{c}} = \frac{a \cdot c}{b}

= \frac{( \frac{c o s ( x )}{s i n ( x )} ) ^{2} \cdot 7 ( \frac{1}{c o s ( x )} ) ^{2} sin ( x )}{1}

(\frac{cos ( x )}{sin ( x )})^{2} = \frac{cos ^{2} ( x )}{sin ^{2} ( x )}

(\frac{cos ( x )}{sin ( x )})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{cos ^{2} ( x )}{sin ^{2} ( x )}

= \frac{7 ( \frac{1}{c o s ( x )} ) ^{2} \frac{c o s ^{2} ( x )}{s i n ^{2} ( x )} sin ( x )}{1}

(\frac{1}{cos ( x )})^{2} = \frac{1}{cos ^{2} ( x )}

(\frac{1}{cos ( x )})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 ^{2}}{cos ^{2} ( x )}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{cos ^{2} ( x )}

= \frac{7 \cdot \frac{c o s ^{2} ( x )}{s i n ^{2} ( x )} \cdot \frac{1}{c o s ^{2} ( x )} sin ( x )}{1}

Multiply

\frac{cos ^{2} ( x )}{sin ^{2} ( x )} \cdot 7 \cdot \frac{1}{cos ^{2} ( x )} sin (x) : \frac{7}{sin ( x )}

\frac{cos ^{2} ( x )}{sin ^{2} ( x )} \cdot 7 \cdot \frac{1}{cos ^{2} ( x )} sin (x)

Multiply fractions:

a \cdot \frac{b}{c} \cdot \frac{d}{e} = \frac{a \cdot b \cdot d}{c \cdot e}

= \frac{cos ^{2} ( x ) \cdot 1 \cdot 7 sin ( x )}{sin ^{2} ( x ) cos ^{2} ( x )}

Cancel the common factor:

cos^{2} (x)

= \frac{1 \cdot 7 sin ( x )}{sin ^{2} ( x )}

Multiply the numbers:

1 \cdot 7 = 7

= \frac{7 sin ( x )}{sin ^{2} ( x )}

Cancel the common factor:

sin (x)

= \frac{7}{sin ( x )}

= \frac{\frac{7}{s i n ( x )}}{1}

Apply the fraction rule:

\frac{a}{1} = a

= \frac{7}{sin ( x )}

= \frac{7}{sin ( x )}

= \frac{7}{sin ( x )}

Manipulating right side

7 tan (x) cos (x) csc^{2} (x)

Express with sin, cos

7 cos (x) csc^{2} (x) tan (x)

Use the basic trigonometric identity:

csc (x) = \frac{1}{sin ( x )}

= 7 cos (x) (\frac{1}{sin ( x )})^{2} tan (x)

Use the basic trigonometric identity:

tan (x) = \frac{sin ( x )}{cos ( x )}

= 7 cos (x) (\frac{1}{sin ( x )})^{2} \frac{sin ( x )}{cos ( x )}

Simplify

7 cos (x) (\frac{1}{sin ( x )})^{2} \frac{sin ( x )}{cos ( x )} : \frac{7}{sin ( x )}

7 cos (x) (\frac{1}{sin ( x )})^{2} \frac{sin ( x )}{cos ( x )}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{sin ( x ) \cdot 7 cos ( x ) ( \frac{1}{s i n ( x )} ) ^{2}}{cos ( x )}

Cancel the common factor:

cos (x)

= sin (x) \cdot 7 (\frac{1}{sin ( x )})^{2}

(\frac{1}{sin ( x )})^{2} = \frac{1}{sin ^{2} ( x )}

(\frac{1}{sin ( x )})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 ^{2}}{sin ^{2} ( x )}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{sin ^{2} ( x )}

= 7 \cdot \frac{1}{sin ^{2} ( x )} sin (x)

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot sin ( x ) \cdot 7}{sin ^{2} ( x )}

Multiply the numbers:

1 \cdot 7 = 7

= \frac{7 sin ( x )}{sin ^{2} ( x )}

Cancel the common factor:

sin (x)

= \frac{7}{sin ( x )}

= \frac{7}{sin ( x )}

= \frac{7}{sin ( x )}

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

Is 7(cot^2(x))/(csc(x))sec^2(x)=7tan(x)cos(x)csc^2(x) ?
The answer to whether 7(cot^2(x))/(csc(x))sec^2(x)=7tan(x)cos(x)csc^2(x) is True